Question

solve.$sqrt{13 - x} = x - 1$select the correct choice below and, if necessary, fill in the answer box to complete your choice.a. the solution(s) is(are) $x = square$(simplify your answer. use a comma to separate answers as needed.)b. there is no solution.

Explanation:

Step1: Square both sides to eliminate root

$$(\sqrt{13-x})^2=(x-1)^2$$
$$13-x=x^2-2x+1$$

Step2: Rearrange to quadratic form

$$x^2-2x+1+x-13=0$$
$$x^2-x-12=0$$

Step3: Factor the quadratic equation

$$(x-4)(x+3)=0$$

Step4: Solve for x values

$x-4=0 \implies x=4$; $x+3=0 \implies x=-3$

Step5: Verify solutions in original equation

For $x=4$: $\sqrt{13-4}=4-1 \implies \sqrt{9}=3 \implies 3=3$, valid.
For $x=-3$: $\sqrt{13-(-3)}=-3-1 \implies \sqrt{16}=-4 \implies 4=-4$, invalid.

Answer:

A. The solution(s) is(are) x = 4